June  2017, 7(2): 201-210. doi: 10.3934/naco.2017014

Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of Sciences, 134, Lermontov St., 664033, Irkutsk, Russia

* Corresponding author: Stepan Sorokin

Received  December 2016 Revised  May 2017 Published  June 2017

Fund Project: This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

We consider a class of rightpoint-constrained state-linear (but non convex) optimal control problems, which takes its origin in the impulsive control framework. The main issue is a strengthening of the Pontryagin Maximum Principle for the addressed problem. Towards this goal, we adapt the approach, based on feedback control variations due to V.A. Dykhta [4,5,6,7]. Our necessary optimality condition, named the feedback maximum principle, is expressed completely in terms of the classical Maximum Principle, but is shown to discard non-optimal extrema. As a connected result, we derive a certain form of duality for the considered problem, and propose the dual version of the proved necessary optimality condition.

Citation: Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
References:
[1]

A. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.   Google Scholar

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.   Google Scholar

[5]

V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57.   Google Scholar

[6]

V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844.  doi: 10.1134/S0005117914050038.  Google Scholar

[7]

V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921.  doi: 10.1134/S0005117914110022.  Google Scholar

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.  Google Scholar

[9]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.  Google Scholar

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.  Google Scholar

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980. Google Scholar

[14]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[15]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.   Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.  Google Scholar

[17]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205.   Google Scholar

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.  Google Scholar

[19]

S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564.  doi: 10.1134/S0005117914090021.  Google Scholar

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003. Google Scholar

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.   Google Scholar

[22]

S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar

show all references

References:
[1]

A. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.   Google Scholar

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.   Google Scholar

[5]

V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57.   Google Scholar

[6]

V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844.  doi: 10.1134/S0005117914050038.  Google Scholar

[7]

V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921.  doi: 10.1134/S0005117914110022.  Google Scholar

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.  Google Scholar

[9]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.  Google Scholar

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.  Google Scholar

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980. Google Scholar

[14]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[15]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.   Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.  Google Scholar

[17]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205.   Google Scholar

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.  Google Scholar

[19]

S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564.  doi: 10.1134/S0005117914090021.  Google Scholar

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003. Google Scholar

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.   Google Scholar

[22]

S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar

[1]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[2]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[3]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[4]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[5]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[6]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[7]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[8]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[9]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[10]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[11]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[13]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[14]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[15]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[16]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[17]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[18]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[19]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[20]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

 Impact Factor: 

Metrics

  • PDF downloads (80)
  • HTML views (114)
  • Cited by (0)

Other articles
by authors

[Back to Top]